I'm reading Evans PDE, on chapter 8.5 the proof of deformation theorem about the calculus of variation.
On page 504 Evans wrote on the top: "we verify that the map $u\to dist(u,A)+dist(u,B)$ is bounded below by a positive constant on each bounded subset of $H$." Who knows how to check this?
Some of my thought, if there is a subsequence $u_k$ such that the sum tends to $0$. Then we deduce $dist(A,B)=0$, as we know the distance between a close set and a compact set are positive, but the bounded subset of $B$ may not be compact. Any suggestions? Thanks in advance!